Skip to main content

On a Generalisation of Finite T-Groups

Abstract

Let \(\sigma =\{\sigma _i |i\in I\}\) be some partition of all primes \({\mathbb {P}}\) and G a finite group. A subgroup H of G is said to be \(\sigma \)-subnormal in G if there exists a subgroup chain \(H=H_0\le H_1\le \cdots \le H_n=G\) such that either \(H_{i-1}\) is normal in \(H_i\) or \(H_i/(H_{i-1})_{H_i}\) is a finite \(\sigma _j\)-group for some \(j \in I\) for \(i = 1, \ldots , n\). We call a finite group G a \(T_{\sigma }\)-group if every \(\sigma \)-subnormal subgroup is normal in G. In this paper, we analyse the structure of the \(T_{\sigma }\)-groups and give some characterisations of the \(T_{\sigma }\)-groups.

This is a preview of subscription content, access via your institution.

References

  1. Al-Sharo, KhA, Skiba, A.N.: On finite groups with \(\sigma \)-subnormal Schmidt subgroups. Commun. Algebra 45, 4158–4165 (2017)

    MathSciNet  Article  Google Scholar 

  2. Ballester-Bolinches, A., Esteban-Romero, R., Asaad, M.: Products of Finite Groups. Walter de Gruyter, Berlin (2010)

    Book  Google Scholar 

  3. Beidleman, J.C., Skiba, A.N.: On \(\tau _{\sigma }\)-quasinormal subgroups of finite groups. J. Group Theory 20(5), 955–964 (2017)

    MathSciNet  Article  Google Scholar 

  4. Doerk, K., Hawkes, T.: Finite Soluble Groups. Walter de Gruyter, Berlin (1992)

    Book  Google Scholar 

  5. Gaschütz, W.: Gruppen, in denen das Normalteilersein transitivist. J. Reine Angew. Math. 198, 87–92 (1957)

    MathSciNet  MATH  Google Scholar 

  6. Guo, W.: Structure Theory for Canonical Classes of Finite Groups. Springer, Heidelberg (2015)

    Book  Google Scholar 

  7. Guo, W., Skiba, A.N.: Finite groups with permutable complete Wielandt sets of subgroups. J. Group Theory 18, 191–200 (2015)

    MathSciNet  Article  Google Scholar 

  8. Guo, W., Skiba, A.N.: Groups with maximal subgroups of Sylow subgroups \(\sigma \)-permutably embedded. J. Group Theory 20(1), 169–183 (2017)

    MathSciNet  Article  Google Scholar 

  9. Guo, W., Skiba, A.N.: On the lattice of \(\Pi _{{\mathfrak{I}}}\)-subnormal subgroups of a finite group. Bull. Aust. Math. Soc. 96(2), 233–244 (2017)

    MathSciNet  Article  Google Scholar 

  10. Guo, W., Skiba, A.N.: Finite groups whose \(n\)-maximal subgroups are \(\sigma \)-subnormal. Sci. China Math. 62(7), 1355–1372 (2019)

    MathSciNet  Article  Google Scholar 

  11. Guo, W., Zhang, C., Skiba, A.N.: On \({\sigma }\)-supersoluble groups and one generalization of \(CLT\)-groups. J. Algebra 512, 92–108 (2018)

    MathSciNet  Article  Google Scholar 

  12. Hall, P.: Theorem like Sylow’s. Proc. Lond. Math. Soc. 6(3), 286–304 (1956)

    MathSciNet  Article  Google Scholar 

  13. Huang, J., Hu, B., Skiba, A.N.: A generalisation of finite \(PT\)-groups. Bull. Aust. Math. Soc. 97(3), 396–405 (2018)

    MathSciNet  Article  Google Scholar 

  14. Peng, T.A.: Finte groups with pronormal subgroups. Proc. Am. Math. Soc. 20, 232–234 (1969)

    Article  Google Scholar 

  15. Peng, T.A.: Pronormality in finite groups. J. Lond. Math. Soc. 3(2), 301–306 (1971)

    MathSciNet  Article  Google Scholar 

  16. Robinson, D.J.S.: A Course in the Theory of Groups. Springer, Heidelberg (1982)

    Book  Google Scholar 

  17. Robinson, D.J.S.: A note on finite groups in which normality is transitive. Proc. Am. Math. Soc. 19, 933–937 (1968)

    MathSciNet  Article  Google Scholar 

  18. Shemetkov, L.A.: Formation of Finite Groups. Nauka, Main Editorial Board for Physical and Mathematical Literature, Moscow (1978)

  19. Skiba, A.N.: On \(\sigma \)-subnormal and \(\sigma \)-permutable subgroups of finite groups. J. Algebra 436, 1–16 (2015)

    MathSciNet  Article  Google Scholar 

  20. Skiba, A.N.: Some characterizations of finite \(\sigma \)-soluble \(P\sigma T\)-groups. J. Algebra 495, 114–129 (2018)

    MathSciNet  Article  Google Scholar 

  21. Skiba, A.N.: On sublattices of the subgroup lattice defined by formation Fitting sets. J. Algebra 550, 69–85 (2020)

    MathSciNet  Article  Google Scholar 

  22. Skiba, A.N.: A generalization of a Hall theorem. J. Algebra Appl. 15(4), 21–36 (2015)

    MathSciNet  Google Scholar 

  23. Skiba, A.N.: On some results in the theory of finite partially soluble groups. Commun. Math. Stat. 4, 281–309 (2016)

    MathSciNet  Article  Google Scholar 

  24. Zhang, C., Skiba, A.N.: On \(\sum _t^{\sigma }\)-closed classes of finite groups. Ukr. Math. J. 70(12), 1707–1716 (2018)

    MathSciNet  Google Scholar 

  25. Zhang, C., Safonov, V.G., Skiba, A.N.: On \(n\)-multiply \(\sigma \)-local formations of finite groups. Commun. Algebra 47(3), 957–968 (2019)

    MathSciNet  Article  Google Scholar 

  26. Zhang, C., Wu, Z., Guo, W.: On weakly \(\sigma \)-permutable subgroups of finite groups. Publ. Math. Debrecen 91, 489–502 (2017)

    MathSciNet  Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Chi Zhang.

Additional information

Research was supported by the Fundamental Research Funds for the Central Universities (No. 2020QN20) and NSFC of China(No. 12001526)

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Zhang, C., Guo, W. & Liu, AM. On a Generalisation of Finite T-Groups. Commun. Math. Stat. 10, 153–162 (2022). https://doi.org/10.1007/s40304-021-00240-z

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40304-021-00240-z

Keywords

  • Finite groups
  • \(\sigma \)-groups
  • Generalised T-groups
  • \(\sigma \)-subnormal
  • The condition \({\mathfrak {R}}_{\sigma _i}\)

Mathematics Subject Classification

  • 20D10
  • 20D15
  • 20D20
  • 20D35